A less-than-truckload carrier collaboration planning problem under dynamic capacities

The author describes a formulation of National Airspace System (NAS) traffic management problem as a multicommodity minimum cost flow problem over a network in space-time. It is a formulation of the multidestination delay minimization problem of managing traffic in the NAS. The model's potential for improving over present practice is described in terms of six assumptions of present practice. The space-time network model can address uncertainty in capacity values, restrictive airways capacities, and hubbing. The minimum cost flow formulation further assumes that attempted entries into the system are deterministic, not Poisson, and that actual entries are controllable. The formulation specifically allows dynamic capacities and possesses good potential for short execution time. >

This paper addresses a single-carrier collaboration problem (SCCP) in which a less-than-truckload (LTL) carrier of interest seeks to collaborate with other carriers by acquiring capacity to service excess demand. The SCCP is addressed from a static (planning) perspective to gain insights into the potential of the collaboration concept for carriers, and its ability to alleviate the effects of increased fuel prices. The study also explores the impact of the degree of collaboration represented by the collaborative discount rate on the carrier of interest. The collaborative strategies are compared to the non-collaboration option represented by a short-term leasing strategy, and the relative benefits of collaboration are computed. Single- and multiple-product SCCPs are formulated as binary (0–1) multi-commodity minimum cost flow problems and are solved using the branch-and-cut algorithm. Experiments are conducted for two transfer cost policies to illustrate insights into the computational performance under varying factors, the effects of different degrees of collaboration, and the impacts of energy costs on the potential for collaboration. The results illustrate that a higher degree of collaboration leads to increased benefits for the carrier of interest and reduced dead-heading for the collaborating carriers. Collaboration also can be critical for the survival of the small- to medium-sized LTL carriers as energy prices escalate given the small industry-wide profit margins.

In this work, we contribute a parallel implementation of the network simplex algorithm that is used for the solution of minimum cost flow problem. In the network simplex algorithm, finding an entering arc requires searching through many arcs to decide which one should be included in the spanning tree solution on the next iteration. We propose finding the entering arc in parallel as it often takes the majority of the execution time. A usual strategy is to pick the arc violating the optimality the most out of all possible candidates. Scanning all arcs can take quite some time, so it is common to consider only a fixed number of arcs which is referred as the block search pivoting rule. Arc scans can easily be done in parallel to find the best candidate as the calculations are independent of each other. We used shared memory parallelism using OpenMP along with vectorization using AVX instructions. We also tried adjusting block sizes to increase the parallel portion of the algorithm. Our dataset consists of various natural and synthetic graphs with sizes up to a billion arc. Our experiments show speedups up to four are possible, though they are typically lower.

A primal partitioning approach for single and non-simultaneous multicommodity flow problems

Methods for assigning a temporary storage inventory, which arrives dynamically at storage areas, to storage space are discussed. It was assumed that no storage location was dedicated to a single type of inventory and that inventory positioned at a storage area could be relocated to another storage area. An attempt was made to minimize the total transportation, storage and handling costs of inventory for a given schedule of arrivals and departures. It is shown that the space allocation problem can be formulated as a minimum cost multi-commodity flow problem. A subgradient optimization algorithm was applied to solve the problem. A numerical experiment was conducted to evaluate the performance of the algorithm.

Decomposition based approaches are recalled from primal and dual point of view. The possibility of building partially disaggregated reduced master problems is investigated. This extends the idea of aggregated-versus-disaggregated formulation to a gradual choice of alternative level of aggregation. Partial aggregation is applied to the linear multicommodity minimum cost flow problem. The possibility of having only partially aggregated bundles opens a wide range of alternatives with different trade-offs between the number of iterations and the required computation for solving it. This trade-off is explored for several sets of instances and the results are compared with the ones obtained by directly solving the natural node-arc formulation. An iterative solution process to the route assignment problem is proposed, based on the well-known Frank Wolfe algorithm. In order to provide a first feasible solution to the Frank Wolfe algorithm, a linear multicommodity min-cost flow problem is solved to optimality by using the decomposition techniques mentioned above. Solutions of this problem are useful for network orientation and design, especially in relation with public transportation systems as the Personal Rapid Transit. A single-commodity robust network design problem is addressed. In this, an undirected graph with edge costs is given together with a discrete set of balance matrices, representing different supply/demand scenarios. The goal is to determine the minimum cost installation of capacities on the edges such that the flow exchange is feasible for every scenario. A set of new instances that are computationally hard for the natural flow formulation are solved by means of a new heuristic algorithm. Finally, an efficient decomposition-based heuristic approach for a large scale stochastic unit commitment problem is presented. The addressed real-world stochastic problem employs at its core a deterministic unit commitment planning model developed by the California Independent System Operator (ISO).

Developing a polynomial time primal network simplex algorithm for the minimum cost flow problem has been a long standing open problem. In this paper, we develop one such algorithm that runs in O(min(n 2m lognC, n 2m2 logn)) time, wheren is the number of nodes in the network,m is the number of arcs, andC denotes the maximum absolute arc costs if arc costs are integer and ∞ otherwise. We first introduce a pseudopolynomial variant of the network simplex algorithm called the “premultiplier algorithm”. We then develop a cost-scaling version of the premultiplier algorithm that solves the minimum cost flow problem in O(min(nm lognC, nm 2 logn)) pivots. With certain simple data structures, the average time per pivot can be shown to be O(n). We also show that the diameter of the network polytope is O(nm logn).

Traffic management and route guidance are optimization problems by nature. In this article, we consider algorithms for centralized route guidance and discuss fairness aspects for the individual user resulting from optimal route guidance policies. The first part of this article deals with the mathematical aspects of these optimization problems from the viewpoint of network flow theory. We present algorithms which solve the constrained multicommodity minimum cost flow problem (CMCF) to optimality. A feasible routing is given by a flow x, and the cost of flow x is the total travel time spent in the network. The corresponding optimum is a restricted system optimum with a globally controlled constrained or fairness factor L > 1. This approach implements a compromise between user equilibrium and system optimum. The goal is to find a route guidance strategy which minimizes global and community criteria with individual needs as constraints. The fairness factor L restricts the set of all feasible routes to the subset of acceptable routes. This might include the avoidance of routes which are much longer than shortest routes, the exclusion of certain streets, preferences for scenic paths, or restrictions on the number of turns to be taken. Most remarkably is that the subset of acceptable routes can also be interpreted as a mental map of routes. In the second part we apply our CMCF algorithms in a large scale multi-agent transportation simulation toolkit, which is called MATSIM-T. We use as initial routes the ones computed by our CMCF algorithms. This choice of initial routes makes it possible to exploit the optimization potential within the simulation much better then it was done before. The result is a speed up of the iteration process in the simulation. We compare the existing simulation toolkit with the new integration of CMCF to proof our results.

A cycle augmentation algorithm for minimum cost multicommodity flows on a ring

The multicommodity flow problem is a generalization of the minimum cost network flow problem. In this problem, several commodities governed by their own network flow constraints share the same underlying network. Given the supplies and demands of the different commodities, the shared capacity of each arc in the network, and cost of flow of each commodity on each arc, the objective is to determine the flow that minimizes the total cost. The multicommodity flow problem has been extensively applied to solve problems in areas such as transportation, telecommunications, and scheduling. In this article, we introduce the multicommodity flow problem, outline some of its applications, and describe efficient methods to solve the problem.

Chapter 1 Applications of network optimization

A problem of finding multicommodity minimum-cost flows in tree-like networks is considered. To solve this problem, an algorithm based on the reduction to a problem of finding a single-commodity flow in a network of arbitrary structure is proposed. For single-commodity flows in tree-like networks, an algorithm using a procedure of border reduction is used.

What makes network architecture efficient? And how do we measure efficiency? In this paper, we study architectural issues in the context of the sensor reachback problem, from an information theoretic perspective. Specifically, we find that in an information-theoretically optimal reachback network, all of the following statements hold: 1) There exists a solution to the problem of transporting the sources over the channels if and only if a suitably defined multicommodity flow is feasible. 2) If a solution exists, then a solution exists based on separate source and channel coding only. 3) When multiple solutions exist, under a natural linear cost model defined in terms of Shannon information, an optimal solution is given by a minimum-cost multicommodity flow. Based on these results we can make a number of statements about what constitutes optimal system architecture for an important class of communication networks, where optimality is defined in a pure information theoretic sense, but has a very clear and intuitive network flow interpretation.

This paper investigates the potential benefits of collaboration among less than truckload (LTL) carriers in fulfilling pickup and delivery jobs. It adopts the commonly used centralized collaborative planning scheme where a central authority pools all jobs and allocate them to the carriers with the objective of minimizing the total transportation cost. However, in this study, LTL carriers are allowed to retain some of the jobs. This extension to the centralized planning scheme is necessary to make collaboration practical. When the carriers are allowed to retain some of their jobs, the allocation of the pooled jobs by the central authority is more complex because it needs to consider the location of the retained jobs. To this end, a mathematical model and a solution method based on large neighborhood search (LNS) are proposed. The experimental results from hypothetical networks provide several important insights regarding cost savings for various collaboration scenarios.

In this paper, we present a new strongly polynomial time algorithm for the minimum cost flow problem, based on a refinement of the Edmonds-Karp scaling technique. Our algorithm solves the uncapacitated minimum cost flow problem as a sequence of O(n log n) shortest path problems on networks with n nodes and m arcs and runs in O(n log n(m + n log n)) time. Using a standard transformation, this approach yields an O(m log n(m + n log n)) algorithm for the capacitated minimum cost flow problem. This algorithm improves the best previous strongly polynomial time algorithm, due to Z. Galil and E. Tardos, by a factor of n2/m. Our algorithm for the capacitated minimum cost flow problem is even more efficient if the number of arcs with finite upper bounds, say m′, is much less than m. In this case, the running time of the algorithm is O((m′ + n) log n(m + n log n)).